Estimating Wave Direction by Using Terrestrial 2 GNSS Reflectometry

: The signal-to-noise ratio (SNR) data is part of the global navigation satellite systems 11 (GNSS) observables. In a marine environment, the oscillation of the SNR data can be used to derive 12 reflector heights. Since the attenuation of the SNR oscillation is related to the roughness of the sea 13 surface, the significant wave height (SWH) of the water surface can be calculated from the analysis 14 of the attenuation. The attenuation depends additionally on the relation between the coherent and 15 the incoherent part of the scattered power. The latter is a function of the correlation length of the 16 surface waves. Because the correlation length changes with respect to the direction of the line of 17 sight relative to the wave direction, the attenuation must show an anisotropic characteristic. In this 18 work, we present a method to derive the wave direction from the anisotropy of the attenuation of 19 the SNR data. The method is investigated based on simulated data as well by the analysis of 20 experimental data from a GNSS station in the North Sea.


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Since more than 20 years reflected GNSS signals are used as a tool to observe diverse 25 environmental conditions by estimating the properties of the reflecting surface. In 1993, Martin-Neira 26 [1] first proposed to use GNSS reflectometry (GNSS-R) in ocean altimetry. Since then, many 27 applications of GNSS-R were developed, reaching from satellite-based sea surface height (SSH)

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Although all GNSS observables are influenced by multipath and can be used for the estimation 34 of the reflector height [7], IPT relates mostly on the analysis of the oscillating structure of the SNR 35 data because it is less influenced by cycle slips or atmospheric refraction as code or carrier phase data. model describes the mean scattered field as a combination of a coherent and an incoherent part. While 48 both parts depend on the roughness of the water surface, the incoherent part depends additionally 49 on the correlation length of the surface. If the correlation length increases, the incoherent part 50 increases, too. That can, as well as an increased roughness, lead to a domination of the incoherent 51 part and yield a loss of coherence.

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Up to now, in the SNR data analysis it was assumed that the correlation length of a sea surface 53 is isotropic. However, a simple gedankenexperiment shows that the correlation length must depend 54 on the wave direction and the direction from which the reflection comes: Consider a simple plane 55 infinite wind-driven wave and define the direction from where the wave comes as the up-wind 56 direction and its opposite as the down-wind direction. The direction perpendicular to the wave front 57 is than referred to as the cross-wind direction. If we intersect the surface in the up-wind or down-58 wind direction and calculate the autocorrelation of the wave heights along the intersection, we will 59 find a correlation length that is related to the wavelength. If we intersect the surface in any off-up-60 wind or off-down-wind direction, the wavelength along the intersection will become longer due to 61 the geometrical stretching of the wave height distribution. If the direction of the intersection tends to 62 the cross-wind direction, the correlation length tends to become infinitely long. Although this model 63 is far too simple, it shows that it should be possible to derive the direction of waves from the 64 anisotropy of εcoh.

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The aim of this work is to demonstrate the possibility to derive the direction of waves from the 66 analysis of GNSS SNR data. In section 2, the basic theory of SNR data analysis and the scattered 67 reflection model from Beckmann and Spizzichino is explained and discussed. Section 3 presents an 68 investigation of the suggested method based on simulations of wave fields. In section 4, real data 69 from a GNSS-equipped tide gauge in the North Sea is analysed and compared to data from a wave 70 buoy and observations of wind directions. Section 5 concludes our findings. The analysis in this work bases on the interference of a direct and a reflected signal that creates 74 a characteristic oscillation in the signal-to-noise ratio (SNR). According to the full model from [10], 75 the SNR is a combination of the direct and the reflected power related to the noise power. Under the 76 assumption of a plane reflecting surface, the SNR can be decomposed into a trend and interference 77 fringes that are attenuated with respect to the elevation angle:

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The amplitude at a specific angle ε can be calculated from the attenuation and the amplitude Amp While the angle ε increases, Ampε becomes small in comparison to the noise of the SNR data and 100 above a certain angle, it disappears in the noise. It is assumed that this is the cutoff angle εcoh, at which 101 the coherence is lost. The threshold at which the loss of coherence is assumed is a matter of definition.

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We suppose to use a threshold that is related to the standard deviation of the SNR data σSNR derived 103 from the non-linear least-squares adjustment multiplied by a factor f. Under these assumptions the 104 coherence is lost if The cutoff angle εcoh is therefore deduced from eq. (3) and (4) Here, σh is the standard deviation of the surface heights. For water surfaces, it is assumed to be 126 approximately a quarter of the SWH [19]. Furthermore,  is the wavelength of the GNSS signal, T is 127 the correlation length of the reflecting surface, i is the incidence angle with i = 90°-ε, r is the 128 reflecting angle,r is the azimuth of the reflection, X and Y are the dimensions of the reflecting area

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A into the coordinate direction x and y and k is again the wave number of the GNSS signal. We are 130 interested in the scattered reflection into the direction of a specular reflection. Therefore, the incidence 131 angle is equal to the reflecting angle and the azimuth of the reflection becomes zero. Since vx and vy 132 become zero and D is 1 for that case, eq. (6) simplifies to Similar to [11], the first term in the bracket in eq. (7) governs the coherent part while the second 134 term governs the incoherent part. Hence, if the term becomes larger than the term coh = 1, the incoherent part dominates the mean scattered power. If the 136 relation between incoh and coh reaches a particular value, the corresponding angle ε can be stated as 137 the cutoff angle εcoh. In accordance with [11]we use a value of 1 for the relation in this work.

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The term incoh depends partly on the geometry of the scattered reflection since A is set as the 139 area of the Fresnel zone. This depends on the reflector height and the angle ε, which also governs   propagate into azimuthal directions with variable amplitudes, frequencies and initial phases: Here, ki is the deep water wave number at angular frequency ωi, while g is the gravitational 172 acceleration.  j is the direction of the elementary wave and  ij is a random initial phase. The 173 amplitude i,j A can be derived approximately from a directional wave spectrum as [22] 174 where  i is an increment of  i and  i is that one of  i .

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whereby the main wave direction was set to zero, resulting in a west-to-east down-wind direction.

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Hence,  j in eq. (9) and (13) are the directions of the spread of the wave. The increment  i was 186 set as one-tenth of the range from a minimal and a maximal wave direction  min and  max .

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To provide realistic simulations, the parameters for wave peak period Tp as well as for  min 188 and  max were derived from real data observed over a period of two months at the wave buoy

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The wave field was simulated for a 1x1 m grid with an extent of 1000 m for both x and y direction.

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The range of SWH was set to be between 0.

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The resulting average correlation length were used to calculate the cutoff angle εcoh according to 226 eq. (8) and the explanation from the previous section. Figure 6 shows the cutoff angles for the different   The difference between the maximal and minimal cutoff angle can be related to the minimal 243 cutoff angle. Figure 7 shows the according differences in percent. The differences are more 244 pronounced for increased SWH. This can imply a more significant estimation of the wave direction

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resulting SWH for the period covered by the GNSS data set is presented in Figure 9 (grey line).

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The distance from the APC to tide gauge zero was taken from information provided by BfG. The

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Additional weather data was provided by the DWD from the close-by weather station Alte Weser

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The GNSS data for every satellite was split into ascending and descending tracks. To avoid 277 influences from the shore of the island, only data with elevation angles over 1° were used. The 278 attenuation of the SNR signal of this antenna type shows a strong degradation for elevation angles 279 above about 10° (see Figure 1). Therefore, we restricted the data set for elevation angles below 10°.

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All elevation angles were corrected for atmospheric refraction and curvature of the reflecting surface 281 as mentioned in section 2. To allow for a good coverage of the horizon, we binned the data into 3h 282 time slots. Since the evolution of the sea state is commonly a slow process, this bin size seems to be

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All resulting azimuths were compared to the wave direction from the buoy. The down-wind or 320 up-wind direction was calculated from the azimuths by adding or subtracting 180° so that the 321 differences to the wave direction from the buoy become minimal. We carried out the same calculation 322 for the wind direction that was taken from the mentioned weather data at the Alte Weser Lighthouse.

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The good agreement of the directions is emphasized by the high correlation between the data 331 sets presented in Figure 12. The correlation coefficient between the wave direction from the buoy and   directions. For lager SWH these differences will at least theoretically be significant.

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The findings from the simulations were verified and largely confirmed by the analysis of data 359 from a GNSS station in the North Sea. For significant results, the correlation to the wave direction 360 observed by a buoy is high and on the same level as the correlation between the wind direction and 361 the wave direction. The significance of the results might be improved if a reliable and automated 362 detection of corrupted data would be applicable.

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Data Availability Statement: The GNSS data used in this study are available from German Federal Institute of 365 Hydrology (Bundesanstalt für Gewässerkunde, BfG) but restrictions apply to the availability of these data, which 366 were used under license for the current study, and so are not publicly available. Data are however available from 367 the corresponding author upon reasonable request and with permission of BfG.

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The tide gauge readings used in this study are available from German Federal Waterways and Shipping

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Administration (Wasserstraßen-und Schifffahrtsverwaltung des Bundes, WSV). The data are available from the 370 corresponding author upon reasonable request.

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The weather data used in this study are available from German Weather Service (Deutscher Wetterdienst, 372 DWD). The data are available from the corresponding author upon reasonable request.

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The data from the buoy and the SWH data from the model used in this study are available from German